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| Dynamics of a Predator-prey Model with Delayand Fear Effect |
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| KeyWord:Predator-prey interaction fear effect delay combined effect Hopf bifurcation |
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| Abstract: |
| Recent manipulations on vertebrates showed that the fear of preda-
tors, caused by prey after they perceived predation risk, could reduce the prey's
reproduction greatly. And it's known that predator-prey systems with fear ef-
fect exhibit very rich dynamics. On the other hand, incorporating the time
delay into predator-prey models could also induce instability and oscillations
via Hopf bifurcation. In this paper, we are interested in studying the com-
bined effects of the fear effect and time delay on the dynamics of the classic
Lotka-Volterra predator-prey model. It's shown that the time delay can cause
the stable equilibrium to become unstable, while the fear effect has a stabi-
lizing effect on the equilibrium. In particular, the model loses stability when
the delay varies and then regains its stability when the fear effect is stronger.
At last, by using the normal form theory and center manifold argument, we
derive explicit formulas which determine the stability and direction of periodic
solutions bifurcating from Hopf bifurcation. Numerical simulations are carried
to explain the mathematical conclusions. |
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